Dirichlet series
| L(s) = 1 | − 96·2-s + 128·3-s + 4.60e3·4-s + 5.46e3·5-s − 1.22e4·6-s + 1.35e4·7-s − 1.31e5·8-s + 8.19e3·9-s − 5.24e5·10-s + 6.47e5·11-s + 5.89e5·12-s − 7.42e5·13-s − 1.29e6·14-s + 6.98e5·15-s + 1.57e6·16-s − 7.55e5·17-s − 7.86e5·18-s + 2.51e7·20-s + 1.72e6·21-s − 6.21e7·22-s − 1.50e7·23-s − 1.67e7·24-s + 3.62e7·25-s + 7.13e7·26-s − 1.31e7·27-s + 6.22e7·28-s − 6.70e7·30-s + ⋯ |
| L(s) = 1 | − 3·2-s + 0.526·3-s + 9/2·4-s + 1.74·5-s − 1.58·6-s + 0.803·7-s − 4·8-s + 0.138·9-s − 5.24·10-s + 4.02·11-s + 2.37·12-s − 2.00·13-s − 2.41·14-s + 0.920·15-s + 3/2·16-s − 0.531·17-s − 0.416·18-s + 7.86·20-s + 0.423·21-s − 12.0·22-s − 2.33·23-s − 2.10·24-s + 3.70·25-s + 6.00·26-s − 0.915·27-s + 3.61·28-s − 2.76·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s+5)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(1000000\) = \(2^{6} \cdot 5^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(65782.1\) |
| Root analytic conductor: | \(2.52062\) |
| Motivic weight: | \(10\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 1000000,\ (\ :[5]^{6}),\ 1)\) |
Particular Values
| \(L(\frac{11}{2})\) | \(\approx\) | \(2.879862318\) |
| \(L(\frac12)\) | \(\approx\) | \(2.879862318\) |
| \(L(6)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{5} T + p^{9} T^{2} )^{3} \) |
| 5 | \( 1 - 1092 p T - 51333 p^{3} T^{2} + 27427512 p^{5} T^{3} - 51333 p^{13} T^{4} - 1092 p^{21} T^{5} + p^{30} T^{6} \) | |
| good | 3 | \( 1 - 128 T + 8192 T^{2} + 162104 p^{4} T^{3} - 605033 p^{6} T^{4} + 580090792 p^{6} T^{5} + 48954489632 p^{6} T^{6} + 580090792 p^{16} T^{7} - 605033 p^{26} T^{8} + 162104 p^{34} T^{9} + 8192 p^{40} T^{10} - 128 p^{50} T^{11} + p^{60} T^{12} \) |
| 7 | \( 1 - 13512 T + 91287072 T^{2} + 811508973008 p T^{3} - 1624254429235233 p^{2} T^{4} - 1147627494425017656 p^{3} T^{5} + \)\(11\!\cdots\!88\)\( p^{4} T^{6} - 1147627494425017656 p^{13} T^{7} - 1624254429235233 p^{22} T^{8} + 811508973008 p^{31} T^{9} + 91287072 p^{40} T^{10} - 13512 p^{50} T^{11} + p^{60} T^{12} \) | |
| 11 | \( ( 1 - 323916 T + 86871291855 T^{2} - 14136588581335480 T^{3} + 86871291855 p^{10} T^{4} - 323916 p^{20} T^{5} + p^{30} T^{6} )^{2} \) | |
| 13 | \( 1 + 742902 T + 275951690802 T^{2} + 104807015418799014 T^{3} - \)\(65\!\cdots\!77\)\( T^{4} - \)\(19\!\cdots\!52\)\( T^{5} - \)\(72\!\cdots\!52\)\( T^{6} - \)\(19\!\cdots\!52\)\( p^{10} T^{7} - \)\(65\!\cdots\!77\)\( p^{20} T^{8} + 104807015418799014 p^{30} T^{9} + 275951690802 p^{40} T^{10} + 742902 p^{50} T^{11} + p^{60} T^{12} \) | |
| 17 | \( 1 + 755118 T + 285101596962 T^{2} + 2224663782493800046 T^{3} - \)\(91\!\cdots\!97\)\( T^{4} - \)\(64\!\cdots\!28\)\( T^{5} - \)\(21\!\cdots\!32\)\( T^{6} - \)\(64\!\cdots\!28\)\( p^{10} T^{7} - \)\(91\!\cdots\!97\)\( p^{20} T^{8} + 2224663782493800046 p^{30} T^{9} + 285101596962 p^{40} T^{10} + 755118 p^{50} T^{11} + p^{60} T^{12} \) | |
| 19 | \( 1 - 36166867207206 T^{2} + \)\(54\!\cdots\!15\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!15\)\( p^{20} T^{8} - 36166867207206 p^{40} T^{10} + p^{60} T^{12} \) | |
| 23 | \( 1 + 15052992 T + 113296284076032 T^{2} + \)\(91\!\cdots\!84\)\( T^{3} + \)\(71\!\cdots\!63\)\( T^{4} + \)\(42\!\cdots\!88\)\( T^{5} + \)\(24\!\cdots\!08\)\( T^{6} + \)\(42\!\cdots\!88\)\( p^{10} T^{7} + \)\(71\!\cdots\!63\)\( p^{20} T^{8} + \)\(91\!\cdots\!84\)\( p^{30} T^{9} + 113296284076032 p^{40} T^{10} + 15052992 p^{50} T^{11} + p^{60} T^{12} \) | |
| 29 | \( 1 - 634112330650806 T^{2} + \)\(88\!\cdots\!35\)\( p T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(88\!\cdots\!35\)\( p^{21} T^{8} - 634112330650806 p^{40} T^{10} + p^{60} T^{12} \) | |
| 31 | \( ( 1 - 76923576 T + 4180287219854295 T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + 4180287219854295 p^{10} T^{4} - 76923576 p^{20} T^{5} + p^{30} T^{6} )^{2} \) | |
| 37 | \( 1 + 32574498 T + 530548959976002 T^{2} - \)\(35\!\cdots\!14\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} + \)\(17\!\cdots\!52\)\( T^{5} + \)\(50\!\cdots\!48\)\( T^{6} + \)\(17\!\cdots\!52\)\( p^{10} T^{7} + \)\(13\!\cdots\!23\)\( p^{20} T^{8} - \)\(35\!\cdots\!14\)\( p^{30} T^{9} + 530548959976002 p^{40} T^{10} + 32574498 p^{50} T^{11} + p^{60} T^{12} \) | |
| 41 | \( ( 1 - 113076636 T + 22611520248205335 T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + 22611520248205335 p^{10} T^{4} - 113076636 p^{20} T^{5} + p^{30} T^{6} )^{2} \) | |
| 43 | \( 1 + 63628752 T + 2024309040538752 T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!23\)\( T^{4} - \)\(28\!\cdots\!64\)\( p T^{5} + \)\(11\!\cdots\!52\)\( p^{2} T^{6} - \)\(28\!\cdots\!64\)\( p^{11} T^{7} + \)\(28\!\cdots\!23\)\( p^{20} T^{8} + \)\(11\!\cdots\!64\)\( p^{30} T^{9} + 2024309040538752 p^{40} T^{10} + 63628752 p^{50} T^{11} + p^{60} T^{12} \) | |
| 47 | \( 1 + 19700448 T + 194053825700352 T^{2} - \)\(41\!\cdots\!64\)\( T^{3} + \)\(61\!\cdots\!23\)\( T^{4} + \)\(52\!\cdots\!52\)\( T^{5} + \)\(18\!\cdots\!48\)\( T^{6} + \)\(52\!\cdots\!52\)\( p^{10} T^{7} + \)\(61\!\cdots\!23\)\( p^{20} T^{8} - \)\(41\!\cdots\!64\)\( p^{30} T^{9} + 194053825700352 p^{40} T^{10} + 19700448 p^{50} T^{11} + p^{60} T^{12} \) | |
| 53 | \( 1 + 950001042 T + 451250989900542882 T^{2} + \)\(26\!\cdots\!34\)\( T^{3} + \)\(18\!\cdots\!63\)\( T^{4} + \)\(79\!\cdots\!88\)\( T^{5} + \)\(28\!\cdots\!08\)\( T^{6} + \)\(79\!\cdots\!88\)\( p^{10} T^{7} + \)\(18\!\cdots\!63\)\( p^{20} T^{8} + \)\(26\!\cdots\!34\)\( p^{30} T^{9} + 451250989900542882 p^{40} T^{10} + 950001042 p^{50} T^{11} + p^{60} T^{12} \) | |
| 59 | \( 1 - 1661016969037924806 T^{2} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!15\)\( p^{20} T^{8} - 1661016969037924806 p^{40} T^{10} + p^{60} T^{12} \) | |
| 61 | \( ( 1 - 801992196 T + 1994865806990658975 T^{2} - \)\(98\!\cdots\!60\)\( T^{3} + 1994865806990658975 p^{10} T^{4} - 801992196 p^{20} T^{5} + p^{30} T^{6} )^{2} \) | |
| 67 | \( 1 - 2502647712 T + 3131622785189417472 T^{2} - \)\(49\!\cdots\!44\)\( T^{3} + \)\(47\!\cdots\!83\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!88\)\( T^{6} - \)\(16\!\cdots\!08\)\( p^{10} T^{7} + \)\(47\!\cdots\!83\)\( p^{20} T^{8} - \)\(49\!\cdots\!44\)\( p^{30} T^{9} + 3131622785189417472 p^{40} T^{10} - 2502647712 p^{50} T^{11} + p^{60} T^{12} \) | |
| 71 | \( ( 1 + 3568766904 T + 12497161629854598375 T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + 12497161629854598375 p^{10} T^{4} + 3568766904 p^{20} T^{5} + p^{30} T^{6} )^{2} \) | |
| 73 | \( 1 - 2304462438 T + 2655273564076451922 T^{2} - \)\(10\!\cdots\!06\)\( T^{3} + \)\(46\!\cdots\!83\)\( T^{4} - \)\(57\!\cdots\!92\)\( T^{5} + \)\(64\!\cdots\!88\)\( T^{6} - \)\(57\!\cdots\!92\)\( p^{10} T^{7} + \)\(46\!\cdots\!83\)\( p^{20} T^{8} - \)\(10\!\cdots\!06\)\( p^{30} T^{9} + 2655273564076451922 p^{40} T^{10} - 2304462438 p^{50} T^{11} + p^{60} T^{12} \) | |
| 79 | \( 1 - 53253337181723476806 T^{2} + \)\(12\!\cdots\!15\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( p^{20} T^{8} - 53253337181723476806 p^{40} T^{10} + p^{60} T^{12} \) | |
| 83 | \( 1 + 88180632 T + 3887911929959712 T^{2} - \)\(16\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} + \)\(59\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!68\)\( T^{6} + \)\(59\!\cdots\!28\)\( p^{10} T^{7} + \)\(11\!\cdots\!03\)\( p^{20} T^{8} - \)\(16\!\cdots\!96\)\( p^{30} T^{9} + 3887911929959712 p^{40} T^{10} + 88180632 p^{50} T^{11} + p^{60} T^{12} \) | |
| 89 | \( 1 - \)\(11\!\cdots\!06\)\( T^{2} + \)\(69\!\cdots\!35\)\( p T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + \)\(69\!\cdots\!35\)\( p^{21} T^{8} - \)\(11\!\cdots\!06\)\( p^{40} T^{10} + p^{60} T^{12} \) | |
| 97 | \( 1 - 343104006 p T + 58860179466624018 p^{2} T^{2} - \)\(56\!\cdots\!54\)\( T^{3} + \)\(34\!\cdots\!03\)\( T^{4} - \)\(76\!\cdots\!28\)\( T^{5} - \)\(25\!\cdots\!32\)\( T^{6} - \)\(76\!\cdots\!28\)\( p^{10} T^{7} + \)\(34\!\cdots\!03\)\( p^{20} T^{8} - \)\(56\!\cdots\!54\)\( p^{30} T^{9} + 58860179466624018 p^{42} T^{10} - 343104006 p^{51} T^{11} + p^{60} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.807355161279537174493956339026, −9.541993365971882840179845743944, −9.316681832365865023410814193161, −9.175852401431695852055325977474, −9.050330583660059136178110773157, −8.452448954667566503169682902532, −8.266463606202724761740674957189, −8.008669053882301374415523855542, −7.76571984768246298819813041877, −6.96625061099426196784129441112, −6.87087864509810877173951289852, −6.60473614768392842161960692277, −6.15629448225037116201783672916, −6.15528575952692494364544430999, −5.15307757553946872242606736930, −4.66795959299419327147647536599, −4.23286549249332268627312822473, −4.09718273821030828859048374502, −2.76067015636073612727335851885, −2.67908604221162747913926324764, −1.95920047526599677587289107791, −1.55854944776641502174177713553, −1.44638601905441035155284825520, −0.70157569794573947950110322850, −0.68891804545024653044520043589, 0.68891804545024653044520043589, 0.70157569794573947950110322850, 1.44638601905441035155284825520, 1.55854944776641502174177713553, 1.95920047526599677587289107791, 2.67908604221162747913926324764, 2.76067015636073612727335851885, 4.09718273821030828859048374502, 4.23286549249332268627312822473, 4.66795959299419327147647536599, 5.15307757553946872242606736930, 6.15528575952692494364544430999, 6.15629448225037116201783672916, 6.60473614768392842161960692277, 6.87087864509810877173951289852, 6.96625061099426196784129441112, 7.76571984768246298819813041877, 8.008669053882301374415523855542, 8.266463606202724761740674957189, 8.452448954667566503169682902532, 9.050330583660059136178110773157, 9.175852401431695852055325977474, 9.316681832365865023410814193161, 9.541993365971882840179845743944, 9.807355161279537174493956339026
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.